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Properties

Label 20.0.12141946772...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 4588681^{2}$
Root discriminant $17.96$
Ramified primes $3, 5, 4588681$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T656

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 29, -14, 22, -84, 67, -91, 89, -104, 99, -52, 62, -43, 26, -14, 8, -8, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 2*x^18 - 8*x^17 + 8*x^16 - 14*x^15 + 26*x^14 - 43*x^13 + 62*x^12 - 52*x^11 + 99*x^10 - 104*x^9 + 89*x^8 - 91*x^7 + 67*x^6 - 84*x^5 + 22*x^4 - 14*x^3 + 29*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^20 + 2*x^18 - 8*x^17 + 8*x^16 - 14*x^15 + 26*x^14 - 43*x^13 + 62*x^12 - 52*x^11 + 99*x^10 - 104*x^9 + 89*x^8 - 91*x^7 + 67*x^6 - 84*x^5 + 22*x^4 - 14*x^3 + 29*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 2 x^{18} - 8 x^{17} + 8 x^{16} - 14 x^{15} + 26 x^{14} - 43 x^{13} + 62 x^{12} - 52 x^{11} + 99 x^{10} - 104 x^{9} + 89 x^{8} - 91 x^{7} + 67 x^{6} - 84 x^{5} + 22 x^{4} - 14 x^{3} + 29 x^{2} + 5 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12141946772837571181640625=3^{10}\cdot 5^{10}\cdot 4588681^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.96$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 5, 4588681$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{191} a^{18} - \frac{3}{191} a^{17} + \frac{2}{191} a^{16} + \frac{13}{191} a^{15} - \frac{49}{191} a^{14} + \frac{16}{191} a^{13} + \frac{37}{191} a^{12} + \frac{84}{191} a^{11} + \frac{50}{191} a^{10} - \frac{3}{191} a^{9} + \frac{40}{191} a^{8} - \frac{6}{191} a^{7} - \frac{62}{191} a^{6} - \frac{42}{191} a^{5} - \frac{13}{191} a^{4} - \frac{49}{191} a^{3} + \frac{95}{191} a^{2} - \frac{49}{191} a + \frac{85}{191}$, $\frac{1}{180523634147} a^{19} + \frac{154663165}{180523634147} a^{18} - \frac{75341888346}{180523634147} a^{17} - \frac{70067563523}{180523634147} a^{16} - \frac{81064995799}{180523634147} a^{15} - \frac{39459930435}{180523634147} a^{14} - \frac{54614536926}{180523634147} a^{13} + \frac{28824121382}{180523634147} a^{12} + \frac{37191861315}{180523634147} a^{11} - \frac{61972472098}{180523634147} a^{10} + \frac{71102374300}{180523634147} a^{9} - \frac{64124894279}{180523634147} a^{8} + \frac{12410230890}{180523634147} a^{7} - \frac{11215759363}{180523634147} a^{6} - \frac{89625495690}{180523634147} a^{5} + \frac{46059837803}{180523634147} a^{4} + \frac{4249848363}{180523634147} a^{3} - \frac{2016847770}{5823343037} a^{2} + \frac{4616284336}{180523634147} a - \frac{46443257927}{180523634147}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{40395385}{945149917} a^{19} - \frac{88505642}{945149917} a^{18} + \frac{139728658}{945149917} a^{17} - \frac{316633435}{945149917} a^{16} + \frac{1071470390}{945149917} a^{15} - \frac{1345421404}{945149917} a^{14} + \frac{1304909299}{945149917} a^{13} - \frac{2769479219}{945149917} a^{12} + \frac{4746280372}{945149917} a^{11} - \frac{5289749377}{945149917} a^{10} + \frac{3688176428}{945149917} a^{9} - \frac{3100468798}{945149917} a^{8} + \frac{6895341891}{945149917} a^{7} - \frac{703231179}{945149917} a^{6} - \frac{1569078531}{945149917} a^{5} + \frac{3469524405}{945149917} a^{4} - \frac{819618435}{945149917} a^{3} + \frac{32433735}{30488707} a^{2} - \frac{5100491531}{945149917} a + \frac{36956188}{945149917} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 34456.4237695 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T656:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 70 conjugacy class representatives for t20n656 are not computed
Character table for t20n656 is not computed

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 10.6.14339628125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
5Data not computed
4588681Data not computed